Asymptotically safe gravity is extremely unlikely to work, as even Weinberg, it's creator, understands. It is pursued as an option, since you must exhaust all roads, but we know enough about gravity today to make it an extraordinarily implausible option.
The issue is the lack of holographic scaling in asymptotically safe gravity. It's an ordinary field theory, so it suffers from the near-horizon blow-up in black hole entropy which is common to all local theories of the formation and annihilation of black holes. This divergence was noticed by 't Hooft, as gave birth to the holograpic principle. The only known fix is to pass to an S-matrix theory, to string theory, in which case the formation and evaporation of black holes is consistent. This is the stringent criterion on quantum gravity, all the field theory conditions are far, far weaker.
This principle, the holographic principle, rules out all known approaches to quantum gravity except strings, and is the reason one can have confidence in the correctness of string theory without saying anything else. A one-parameter prediction is unlikely to be signficant evidence for asymptotically safe gravity, especially considering that it is theoretically nearly ruled out.
The analysis in the paper is also not particularly stringent in predicting $126~\text{GeV}$. It's a result of RG running with some ad-hoc assumption on the fixed point. They could have gotten essentially any answer less than $8~\text{TeV}$ by adjusting their assumptions, since they don't deal with the hypothetical strong gravity fixed point very much, beyond getting some guesses for the couplings from it. So this is just a lucky guess.
Asymptotically safe gravity is not really a viable option, only string theory is.
